It makes the chance much smaller than it normally is. According to this post from 2006, turn-generation tooltips say the chance of regenerating creature getting an affliction is 1/8th of non-regenerating one.
According to FAQ, "Having regeneration will reduce your chance of getting a battle affliction by 75%."
The tooltips are more correct. 1/8 is 12.5% so it's a 87% reduction. The FAQ estimate is thus too low. I had to daw something out of the hat because I didn't remember the exact number.